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Chapter 1: Problem 50

Solve the formula for the specified variable. $$C D+C=P C+R \text { for } C$$

Short Answer

Expert verified
\(C = \frac{R}{D + 1 - P}\)

Step by step solution

01

Arrange All Terms with 'C' on One Side

The given equation is \( CD + C = PC + R \). Start by rearranging the terms so that all terms involving the variable \( C \) are on one side of the equation. To do this, subtract \( PC \) from both sides: \[ CD + C - PC = R \].
02

Factor Out the Common Variable 'C'

Notice that \( C \) is common in the terms on the left side \( CD + C - PC \). Factor \( C \) out of these terms: \[ C(D + 1 - P) = R \].
03

Solve for 'C'

To isolate \( C \), divide both sides of the equation by the expression \( (D + 1 - P) \): \[ C = \frac{R}{D + 1 - P} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Isolation
When we solve for a specific variable in an equation, the process is known as "Variable Isolation". In simple terms, this means rearranging the equation to get the variable by itself on one side of the equation.
In the given equation, \( CD + C = PC + R \), the goal is to solve for \( C \). This involves moving all terms with \( C \) on one side. This gives us the chance to see how \( C \) interacts with other terms and provides a way to isolate it.
  • Start with the equation and identify terms with the variable of interest, \( C \) here.
  • Subtract or add terms from both sides to move all instances of the variable to one side of the equation.
Here, subtract \( PC \) from both sides to gather all \( C \) terms on the left, resulting in \( CD + C - PC = R \). This setup is crucial because once all terms with \( C \) are on one side, you can proceed to the next step with clarity and purpose.
Factoring
Factoring is a technique used to simplify expressions, which helps when isolating a variable. In the equation \( CD + C - PC = R \), all terms with \( C \) are on one side, making it possible to factor \( C \) out.
  • Identify common factors in the expression on one side of the equation.
  • Factor out the common element, in this case, \( C \) from \( CD + C - PC \).
After identifying that \( C \) is a common factor, rewrite the left side as \( C(D + 1 - P) \). Factoring like this simplifies the expression and sets up the equation for easier manipulation. It reduces complexity by turning multiple terms into a single term, making it straightforward to handle in subsequent steps.
Algebraic Manipulation
Once the equation is factored, algebraic manipulation helps in isolating the variable completely. Now you have \( C(D + 1 - P) = R \). To solve for \( C \), we need to get \( C \) alone.
  • When a variable is a factor in a product, divide both sides by the other factor to isolate the variable.
  • Ensure the expression in the denominator is not zero since division by zero is undefined.
In our equation, divide both sides by \( (D + 1 - P) \) to retrieve \( C = \frac{R}{D + 1 - P} \). This manipulation ensures that \( C \) is isolated, with uncomplicated terms on the other side. The solution underlines understanding and systematically reduces an equation to find the desired variable neatly. Here, the careful execution of algebraic manipulation leads directly to the satisfying solution for \( C \).

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Most popular questions from this chapter

Choose the equation that best describes the table of data. $$\begin{array}{|c|c|}\hline \boldsymbol{x} & \boldsymbol{y} \\\\\hline 1 & 0.8 \\\2 & -0.4 \\\3 & -1.6 \\\4 & -2.8 \\\5 & -4.0 \\\\\hline\end{array}$$ (1) \(y=-1.2 x+2\) (2) \(y=-1.2 x^{2}+2\) (3) \(y=0.8 \sqrt{x}\) (4) \(y=x^{3 / 4}-0.2\)

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[5]{\frac{5 x^{7} y^{2}}{8 x^{3}}}$$

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt{5 x y^{2}} \sqrt{15 x^{3} y^{3}}$$

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[4]{\frac{5 x^{8} y^{3}}{27 x^{2}}}$$

Simplify the expression. $$\frac{(6 x+1)^{3}\left(27 x^{2}+2\right)-\left(9 x^{3}+2 x\right)(3)(6 x+1)^{2}(6)}{(6 x+1)^{6}}$$
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